Assouad and lower dimensions of graph-directed Bedford–McMullen carpets

Abstract

We calculate the Assouad and the lower dimensions of graph-directed Bedford–McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrast to known results on Hausdorff and box dimensions. We also investigate the relationship between distinct dimensions. In particular, we identify an equivalent condition when the box and the Assouad dimensions coincide, and show that under this condition, the Hausdorff dimension attains the same value.